Algebraic and Affine Pattern Avoidance

TL;DR

This paper explores the algebraic structures related to pattern avoidance in permutations, establishing new characterizations and extending these concepts to affine settings, thereby enriching the theoretical framework of combinatorial pattern analysis.

Contribution

It introduces algebraic characterizations of pattern containment and extends these concepts to affine permutations, providing new tools for pattern avoidance analysis.

Findings

Characterization of pattern containment via factorizations.

Extension of algebraic pattern avoidance tools to affine permutations.

New connections between 0-Hecke monoid, Catalan monoid, and pattern avoidance.

Abstract

We investigate various connections between the 0-Hecke monoid, Catalan monoid, and pattern avoidance in permutations, providing new tools for approaching pattern avoidance in an algebraic framework. In particular, we characterize containment of a class of `long' patterns as equivalent to the existence of a corresponding factorization. We then generalize some of our constructions to the affine setting.